STRATEGY FOR GRAPHING TRIGONOMETRIC FUNCTIONS
USING
MAPPING & SUPERPOSITION
Dr. Marwan Zabdawi
Gordon State College
mzabdawi@gordonstate.edu
81 Chad Court
McDonough, GA 30253
Abstract
Virtually all books in Pre-Calc. do not give the student an outline on how to graph trigonometric
functions with amplitudes, non-standard periods, phase shifts and vertical translation. Instead,
examples are given for specific scenarios by fixing the scale and changing the window/graph.
Hence, the student is left bewildered in a mathematical maze trying to find a way out. This paper
uses the concepts of mapping and superposition to resolve any combination of scenarios and even
all scenarios combined. Mapping and superposition is done by fixing the window/graph and
changing the scale. Two examples will be tabulated and graphed with a strategy where the
student can go through it mechanically and without a hitch. The TI-84 Plus will be used for
verification of two examples in the form:
𝒚 = 𝒂𝑺𝒆𝒄𝒃(𝒙 ± 𝒄) ± 𝒅 and 𝒚 = 𝒂𝑻𝒂𝒏𝒃(𝒙 ± 𝒄) ± 𝒅
A bullet format outline follows with two examples to demonstrate the procedure.
Procedure:
1. For graphing functions in the format:
𝒚 = 𝒂𝑺𝒆𝒄𝒃(𝒙 ± 𝒄) ± 𝒅, we need to graph 𝒚 = 𝒂𝑪𝒐𝒔𝒃(𝒙 ± 𝒄) ± 𝒅
a)
Amplitude = |𝑎|
2𝜋
𝑏
b)
Period =
c)
+𝑐
Phase Shift = 𝐼𝑓 (
−𝑐
d)
𝐺𝑟𝑎𝑝ℎ 𝑖𝑠 𝑠ℎ𝑖𝑓𝑡𝑒𝑑 𝑐 𝑢𝑛𝑖𝑡𝑠 𝑡𝑜 𝑡ℎ𝑒 𝑙𝑒𝑓𝑡
)
𝐺𝑟𝑎𝑝ℎ 𝑖𝑠 𝑠ℎ𝑖𝑓𝑡𝑒𝑑 𝑐 𝑢𝑛𝑖𝑡𝑠 𝑡𝑜 𝑡ℎ𝑒 𝑟𝑖𝑔ℎ𝑡
+𝑑
𝐺𝑟𝑎𝑝ℎ 𝑖𝑠 𝑚𝑜𝑣𝑒𝑑 𝑑 𝑢𝑛𝑖𝑡𝑠 𝑢𝑝
Vertical Translation = 𝐼𝑓 (
)
− 𝑑 𝐺𝑟𝑎𝑝ℎ 𝑖𝑠 𝑚𝑜𝑣𝑒𝑑 𝑑 𝑢𝑛𝑖𝑡𝑠 𝑑𝑜𝑤𝑛
2. Alignment/Mapping is defined as finding the x-values for which the new signal has the
same y-values as the original classical signal before horizontal shifting, amplifications
and/or vertical translation.
3. Superposition is defined as super-imposing the new graph on the fixed window of the
original classical signal by simply changing the scale.
It sounds complicated, but the following two examples will illustrate the idea, and we’ll
use the acronym ASAUD which stands for (Aligned, Shifted, Amplified, UP or Down).
𝝅
Let us graph 𝒚 = −𝟐𝑺𝒆𝒄𝟐 (𝒙 + 𝟐 ) + 𝟏, … … … 𝒆𝒒. (𝟏) hence, we need to graph
𝝅
𝟐
𝒚 = −𝟐𝑪𝒐𝒔𝟐 (𝒙 + ) + 𝟏, … … … 𝒆𝒒. (𝟐)
a) Amplitude = |−2| = 2
b) Period =
2𝜋
2
=𝜋
c) Phase Shift =
𝝅
𝟐
units to the left (subtract)
d) Vertical Translation = 1 unit up.
The original classical signal which is the building block for this graph is y = Cos(x)
Fig. (1)
The following table with the use of the acronym ASAUD will transform the graph given in Fig.
(1) to the graph of eq. (2).
For the alignment, we divide the new period 𝜋 into four cycles to align it with the four cycles of
the original classical signal y = Cos(x).
x of
y= Cos(x)
Aligns with x
0
0
𝜋
2
𝜋
𝜋
4
𝜋
2
3𝜋
4
𝜋
3𝜋
2
2𝜋
Shifted to x
Y values of
y= Cos(x)
Y Amplified
by -2
Y Moved up
by 1
𝜋
2
𝜋
−
4
0
1
-2
-1
0
0
1
-1
2
3
0
0
1
1
-2
-1
𝜋
−
2
−
𝜋
4
𝜋
2
We now take the last x and y values from the table above to plot eq. (2). Because of the
linearity of the transformations, the amplifications and the vertical translations were done
solely on the original signal y= Cos(x) regardless of the horizontal shifting. The minus
sign in the amplification indicates that the signal had been flipped upside down as:
Now by fixing the window and changing the scale, we simply transform the graph of
𝜋
y=Cos(x) to the graph of 𝑦 = −2𝐶𝑜𝑠2 (𝑥 + 2 ) + 1
Bearing in mind that the axes are inserted after the graph is drawn with its tick marks
taken from the last x and y values from the table above. Keeping track of the wanted
1
graph of eq. (1) and recalling that 𝑆𝑒𝑐𝑥(𝑥) = 𝐶𝑜𝑠(𝑥) , the position of the asymptote of the
secant function is the same as the position of the zero value of the cosine function before
the vertical translation (where the dashed line intersected the graph), and hence the final
graph of eq.(1) is:
Procedure:
4. For graphing functions in the format:
𝒚 = 𝒂𝑻𝒂𝒏𝒃(𝒙 ± 𝒄) ± 𝒅
a)
Amplitude = Infinity
𝜋
b)
Period = 𝑏
c)
d)
+𝑐
Phase Shift = 𝐼𝑓 (
−𝑐
𝐺𝑟𝑎𝑝ℎ 𝑖𝑠 𝑠ℎ𝑖𝑓𝑡𝑒𝑑 𝑐 𝑢𝑛𝑖𝑡𝑠 𝑡𝑜 𝑡ℎ𝑒 𝑙𝑒𝑓𝑡
)
𝐺𝑟𝑎𝑝ℎ 𝑖𝑠 𝑠ℎ𝑖𝑓𝑡𝑒𝑑 𝑐 𝑢𝑛𝑖𝑡𝑠 𝑡𝑜 𝑡ℎ𝑒 𝑟𝑖𝑔ℎ𝑡
+𝑑
𝐺𝑟𝑎𝑝ℎ 𝑖𝑠 𝑚𝑜𝑣𝑒𝑑 𝑑 𝑢𝑛𝑖𝑡𝑠 𝑢𝑝
Vertical Translation = 𝐼𝑓 (
)
− 𝑑 𝐺𝑟𝑎𝑝ℎ 𝑖𝑠 𝑚𝑜𝑣𝑒𝑑 𝑑 𝑢𝑛𝑖𝑡𝑠 𝑑𝑜𝑤𝑛
𝝅
𝟐
Let us graph 𝒚 = −𝟐𝑻𝒂𝒏𝟐 (𝒙 − ) − 𝟏, … … … 𝒆𝒒. (𝟑)
a)
b)
Amplitude = Infinity
𝜋
Period = 2
c)
Phase Shift = to the right (Add)
𝜋
2
d)
Vertical Translation = 1 unit down
The original classical signal which is the building block for this graph is y = Tan(x)
Fig. (2)
The following table with the use of the acronym ASAUD will transform the graph given in Fig.
(2) to the graph of eq. (3).
For the alignment, we divide the new period
𝜋
2
into two cycles to align it with the two cycles of
the original classical signal y = Tan(x).
x of
y= Tan(x)
𝜋
−
2
0
Aligns with x
Shifted to x
Y values of
y= Tan(x)
Y Amplified
by -2
Y Moved
down by 1
𝜋
4
0
-∞
∞
∞
0
0
-1
𝜋
2
𝜋
4
𝜋
4
𝜋
2
3𝜋
4
∞
−∞
−∞
−
𝜋
+
2
We now take the last x and y values from the table above to plot eq. (3). Again, because
of the linearity of the transformations, the amplifications and the vertical translations
were done solely on the original signal y= Tan(x) regardless of the horizontal shifting.
The minus sign in the amplification indicates that the signal had been flipped upside
down as:
Now by fixing the window and changing the scale, we simply transform the graph of
𝜋
y=Tan(x) to the graph of 𝑦 = −2𝑇𝑎𝑛2 (𝑥 − 2 ) − 1
Bearing in mind that the axes are inserted after the graph is drawn with its tick marks
taken from the last x and y values from the table above.
The same procedure can be applied to graphs in the format:
𝒚 = 𝒂𝑪𝒔𝒄𝒃(𝒙 ± 𝒄) ± 𝒅 and 𝒚 = 𝒂𝑪𝒐𝒕𝒃(𝒙 ± 𝒄) ± 𝒅